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 P: 1,666 That's ok Mathwonk. I figured you are busy. Thanks for the Walker reference and the other references above. They're helpful. Also, I'm afraid I used an abuse of notation above. That's really three factors: $$f(z,w)=\left(w-g(z)\right)\left(w-h(\epsilon_1z^{1/2})\right)\left(w-h(\epsilon_2 z^{1/2})\right)$$ however the two factors of $h(z^{1/2})$ come from the same 2-valued manifold, just different single-valued determinations of it. Just thought my condenced notation was cleaner-looking. And the expansion comes from Newton-Puiseux's Theorem: The quotient field $\mathbb{C}((z^*))$ of convergent fractional power series (which I assume includes all analytic functions) is algebraically closed so for $f(z,w)=a_n(z)w^n+\cdots+a_0(z)$, with $a_i(z)\in \mathbb{C}((z^*))$, we have $$f(z,w)=\prod_{i=1}^n \left(w-g_i(z^{1/k_i})\right)$$ with $g_i(z^{1/k_i})\in\mathbb{C}((z^*))$. One reason I knew the expansion was in terms of $g(z)$ and $h(z^{1/2})$, is because I drew it: (Not entirely sure I'm quoting that theorem precisely. This is all very new to me.) Attached Thumbnails